In medical ultrasound, many diagnostic procedures begin from the determination of the power spectrum of the returning signal. Certain of these diagnostic procedures can provide better diagnostic precision if the power spectrum can be derived at the same sampling rate as that of the incoming signal. Attempting to do this real-time spectral analysis with a dedicated computer using known fast Fourier transform procedures has previously been inordinately expensive in terms of digital hardware and of the power consumption by that hardware, when attempting to meet speed requirements. Accordingly, the development of more refined medical diagnoses using ultrasound has been hampered.
The reader is referred to the text THE FAST FOURIER TRANSFORM by E. Oran Brigham, copyright 1974 by Prentice Hall, Inc.., Englewood Cliffs, N.J. The first five chapters of this book describe the continuous Fourier transform and its various properties.
The continuous Fourier transform of a repetitive function in the time domain (repeating at T.sub.0 intervals) comprises a spectrum of discrete frequencies that are harmonics of the repetition frequency F.sub.0 =T.sub.0.sup.-1 and have both real and imaginary components, one of which components may be zero-valued. As described by Brigham in Chapter 6 of his book, a discrete Fourier transform can be generated, which approximates the continuous Fourier transform and facilitate calculation by computer Brigham in subchapter 6-2 indicates that the discrete Fourier transform G(nN/t) of a function g(kt) sampled periodically in time T can be defined as follows. ##EQU1## Note the above definition describes the computation of N equations for describing the N time samples and the N spectral values in one period F.sub.0 of the frequency domain respectively.
If these N equations are arranged in matrix format replacing exp(-j2.pi./N) with the complex number W, it appears at first as if N.sup.2 complex multiplications and N(N-1) complex additions are necessary for carrying out computation. It has been found that where N is a power of two, the total number of multiplications can be reduced to N log.sub.2 N by taking advantage of the fact that certain powers of W are equal to others to allow factoring the large square matrix in powers on W into the product of smaller matrices, after some juggling of the order of its rows and columns. Since the complexity of the computation of the matrix equation is primarily dependent on the number of multiplications, these factoring processes can serve as the basis for computations of discrete Fourier transform that are more efficient overall. These computations are commonly referred to as the "fast Fourier transform" or "FFT".
The computational processes for discrete Fourier transform that are known in the prior art may be characterized as being finite-impulse-response (FIR) digital filtering processes, it is here pointed out FIR digital time-domain filtering processes are ones in which the response to an impulse input actually becomes zero-valued in a finite time interval. In the prior art discrete Fourier transform calculations the truncation interval forms a low-pass filtering kernel of finite width and imposes FIR characteristics on the transform results.
A characteristic of an FIR digital filtering process is that a filter kernel must be applied to each input signal sample. Accordingly, almost invariably FIR filtering is done on a non-recursive basis using a tapped-delay-line structure sometime referred to as a transversal filter. Using such non-recursive filtering to achieve a continuous filter response entails many multiplications. In calculating discrete Fourier transforms in accordance with the prior art, this same requirement that all multiplications must be made respective to each input signal sample obtains, presuming it is attempted to generate the discrete Fourier transform at the same rate F as input signal samples are supplied--that is, with truncation interval continuously sliding in time. To keep the number of multiplications per time period within practical bounds, however, the custom is to step the truncation interval N input signal samples at a time and to calculate the discrete Fourier transform at rate F/N.
Another class of digital time-domain filtering processes that is generally known may be characterized as being infinite impulse response (IIR) character IIR time-domain filtering processes are ones in which the response to an impulse input extends over an infinitude of positive time. While response may diminish to negligible values in extended time, theoretically it never actually goes to zero. IIR filter characteristics normally obtain when recursive filtering processes take place.
Recursive digital filtering uses the previous filter output signal value(s) in computing the current response to an input signal. Recursive digital filtering can reduce the number of digital multiplications required to accomplish many filtering tasks, and this is generally the reason for their use. Recursive digital filters with IIR characteristics have an impulse response that is asymmetric in the time domain, which is commonly viewed as being disadvantageous.